Growing uniform planar maps face by face

Abstract

We provide “growth schemes” for inductively generating uniform random 2p$$ 2p $$ ‐angulations of the sphere with n$$ n $$ faces, as well as uniform random simple triangulations of the sphere with 2n$$ 2n $$ faces. In the case of 2p$$ 2p $$ ‐angulations, we provide a way to insert a new face at a random location in a uniform 2p$$ 2p $$ ‐angulation with n$$ n $$ faces in such a way that the new map is precisely a uniform 2p$$ 2p $$ ‐angulation with n+1$$ n+1 $$ faces. Similarly, given a uniform simple triangulation of the sphere with 2n$$ 2n $$ faces, we describe a way to insert two new adjacent triangles so as to obtain a uniform simple triangulation of the sphere with 2n+2$$ 2n+2 $$ faces. The latter is based on a new bijective presentation of simple triangulations that relies on a construction by Poulalhon and Schaeffer.